In the present proposal we consider the initial value problems of type
\begin{eqnarray*}
\frac{\partial \omega}{\partial t}=L\omega=\sum_{i=0}^{3}A^{(i)}(t,x)\frac{\partial \omega}{\partial_{x_{i}}}+B(t,x)\omega +C(t,x)\\
\omega(0,x)=\Psi(x)
\end{eqnarray*}
in the space of $q$-generalized regular functions in the sense of Quaternionic Analysis satisfying the differential equation
\begin{equation}
\mathfrak{D}=\sum_{j=0}^{3}q_{j}\left[e_{j}\partial_{j}-\lambda_{j}\right]
\end{equation}
where $t \in[0,T]$ is the time variable, $x$ runs in a bounded and simply connected domain in $\Re^{4}$, the $q_{i}$ are considered real functions and $\lambda_{i}$ are considered constants. We prove necessary and sufficient conditions on the coefficients of the operator $L$ under which $L$ is associated with the operator $\mathfrak{D}$. This criterion makes it possible to construct operators $L$ for which the initial value problem is uniquely soluble for an arbitrary initial $q$-generalized regular function $\Psi(x)$ by the method of associated spaces constructed by W. Tutschke (Teubner Leipzig and Springer Verlag, 1989) and the solution is also $q$-generalized regular for each $t$.